LMFDB - Embedded newform 925.2.b.f.149.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [925,2,Mod(149,925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("925.149"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 925 = 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	925.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-20,0,-12,0,0,-12,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$7.38616218697$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.60703296077824.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 20x^{8} + 142x^{6} + 420x^{4} + 457x^{2} + 144 $ x^10 + 20x^8 + 142x^6 + 420x^4 + 457x^2 + 144
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 185)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.10
Root			$-2.72362i$ of defining polynomial
Character	$\chi$	$=$	925.149
Dual form			925.2.b.f.149.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.72362i q^{2} +2.29298i q^{3} -5.41809 q^{4} -6.24519 q^{6} +3.82710i q^{7} -9.30957i q^{8} -2.25774 q^{9} -4.41809 q^{11} -12.4236i q^{12} +3.67583i q^{13} -10.4236 q^{14} +14.5195 q^{16} -2.28688i q^{17} -6.14922i q^{18} +2.39037 q^{19} -8.77545 q^{21} -12.0332i q^{22} +0.265251i q^{23} +21.3466 q^{24} -10.0116 q^{26} +1.70198i q^{27} -20.7356i q^{28} +6.58595 q^{29} +2.34076 q^{31} +20.9265i q^{32} -10.1306i q^{33} +6.22860 q^{34} +12.2326 q^{36} +1.00000i q^{37} +6.51044i q^{38} -8.42859 q^{39} -4.41809 q^{41} -23.9010i q^{42} -7.71249i q^{43} +23.9376 q^{44} -0.722443 q^{46} +10.9285i q^{47} +33.2929i q^{48} -7.64669 q^{49} +5.24377 q^{51} -19.9160i q^{52} +0.109574i q^{53} -4.63555 q^{54} +35.6286 q^{56} +5.48105i q^{57} +17.9376i q^{58} +2.00504 q^{59} +3.96271 q^{61} +6.37534i q^{62} -8.64059i q^{63} -27.9567 q^{64} +27.5918 q^{66} +6.80664i q^{67} +12.3905i q^{68} -0.608215 q^{69} -5.79485 q^{71} +21.0186i q^{72} +0.140654i q^{73} -2.72362 q^{74} -12.9512 q^{76} -16.9085i q^{77} -22.9563i q^{78} +6.62418 q^{79} -10.6758 q^{81} -12.0332i q^{82} -13.9904i q^{83} +47.5462 q^{84} +21.0059 q^{86} +15.1014i q^{87} +41.1305i q^{88} -14.8139 q^{89} -14.0678 q^{91} -1.43716i q^{92} +5.36731i q^{93} -29.7651 q^{94} -47.9839 q^{96} -8.94394i q^{97} -20.8266i q^{98} +9.97490 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 20 q^{4} - 12 q^{6} - 12 q^{9} - 10 q^{11} + 16 q^{14} + 32 q^{16} + 8 q^{19} + 6 q^{21} + 84 q^{24} - 8 q^{26} + 8 q^{29} + 16 q^{31} + 64 q^{34} + 32 q^{36} - 4 q^{39} - 10 q^{41} + 92 q^{44} - 44 q^{49}+ \cdots + 20 q^{99}+O(q^{100}) $ 10 * q - 20 * q^4 - 12 * q^6 - 12 * q^9 - 10 * q^11 + 16 * q^14 + 32 * q^16 + 8 * q^19 + 6 * q^21 + 84 * q^24 - 8 * q^26 + 8 * q^29 + 16 * q^31 + 64 * q^34 + 32 * q^36 - 4 * q^39 - 10 * q^41 + 92 * q^44 - 44 * q^49 - 4 * q^51 + 20 * q^54 + 16 * q^56 + 60 * q^59 - 28 * q^61 - 40 * q^64 + 96 * q^66 - 16 * q^69 - 14 * q^71 - 4 * q^74 + 24 * q^76 - 56 * q^79 - 62 * q^81 + 36 * q^84 + 88 * q^86 - 12 * q^89 - 28 * q^91 - 8 * q^94 - 84 * q^96 + 20 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/925\mathbb{Z}\right)^\times$.

$n$	$76$	$852$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.72362i	1.92589i	0.269701	+	0.962944i	$0.413075\pi$
$2$	2.72362i	1.92589i	−0.269701	+	0.962944i	$0.586925\pi$
$3$	2.29298i	1.32385i	0.749570	+	0.661925i	$0.230260\pi$
$3$	2.29298i	1.32385i	−0.749570	+	0.661925i	$0.769740\pi$
$4$	−5.41809	−2.70905
$5$	0	0
$5$	0	0
$6$	−6.24519	−2.54959
$7$	3.82710i	1.44651i	0.690582	+	0.723254i	$0.257354\pi$
$7$	3.82710i	1.44651i	−0.690582	+	0.723254i	$0.742646\pi$
$8$	− 9.30957i	− 3.29143i
$9$	−2.25774	−0.752580
$10$	0	0
$11$	−4.41809	−1.33210	−0.666052	−	0.745905i	$-0.732017\pi$
$11$	−4.41809	−1.33210	−0.666052	+	0.745905i	$0.732017\pi$
$12$	− 12.4236i	− 3.58637i
$13$	3.67583i	1.01949i	0.860325	+	0.509746i	$0.170261\pi$
$13$	3.67583i	1.01949i	−0.860325	+	0.509746i	$0.829739\pi$
$14$	−10.4236	−2.78581
$15$	0	0
$16$	14.5195	3.62988
$17$	− 2.28688i	− 0.554651i	−0.960776	−	0.277325i	$-0.910552\pi$
$17$	− 2.28688i	− 0.554651i	0.960776	−	0.277325i	$-0.0894480\pi$
$18$	− 6.14922i	− 1.44938i
$19$	2.39037	0.548387	0.274194	−	0.961674i	$-0.411589\pi$
$19$	2.39037	0.548387	0.274194	+	0.961674i	$0.411589\pi$
$20$	0	0
$21$	−8.77545	−1.91496
$22$	− 12.0332i	− 2.56548i
$23$	0.265251i	0.0553087i	0.999618	+	0.0276544i	$0.00880378\pi$
$23$	0.265251i	0.0553087i	−0.999618	+	0.0276544i	$0.991196\pi$
$24$	21.3466	4.35736
$25$	0	0
$26$	−10.0116	−1.96343
$27$	1.70198i	0.327547i
$28$	− 20.7356i	− 3.91865i
$29$	6.58595	1.22298	0.611490	−	0.791252i	$-0.290570\pi$
$29$	6.58595	1.22298	0.611490	+	0.791252i	$0.290570\pi$
$30$	0	0
$31$	2.34076	0.420413	0.210207	−	0.977657i	$-0.432586\pi$
$31$	2.34076	0.420413	0.210207	+	0.977657i	$0.432586\pi$
$32$	20.9265i	3.69931i
$33$	− 10.1306i	− 1.76351i
$34$	6.22860	1.06820
$35$	0	0
$36$	12.2326	2.03877
$37$	1.00000i	0.164399i
$37$	1.00000i	0.164399i
$38$	6.51044i	1.05613i
$39$	−8.42859	−1.34965
$40$	0	0
$41$	−4.41809	−0.689990	−0.344995	−	0.938605i	$-0.612119\pi$
$41$	−4.41809	−0.689990	−0.344995	+	0.938605i	$0.612119\pi$
$42$	− 23.9010i	− 3.68800i
$43$	− 7.71249i	− 1.17614i	−0.808809	−	0.588072i	$-0.799887\pi$
$43$	− 7.71249i	− 1.17614i	0.808809	−	0.588072i	$-0.200113\pi$
$44$	23.9376	3.60873
$45$	0	0
$46$	−0.722443	−0.106518
$47$	10.9285i	1.59409i	0.603920	+	0.797045i	$0.293605\pi$
$47$	10.9285i	1.59409i	−0.603920	+	0.797045i	$0.706395\pi$
$48$	33.2929i	4.80542i
$49$	−7.64669	−1.09238
$50$	0	0
$51$	5.24377	0.734275
$52$	− 19.9160i	− 2.76185i
$53$	0.109574i	0.0150512i	0.999972	+	0.00752559i	$0.00239549\pi$
$53$	0.109574i	0.0150512i	−0.999972	+	0.00752559i	$0.997605\pi$
$54$	−4.63555	−0.630819
$55$	0	0
$56$	35.6286	4.76108
$57$	5.48105i	0.725983i
$58$	17.9376i	2.35532i
$59$	2.00504	0.261034	0.130517	−	0.991446i	$-0.458336\pi$
$59$	2.00504	0.261034	0.130517	+	0.991446i	$0.458336\pi$
$60$	0	0
$61$	3.96271	0.507374	0.253687	−	0.967286i	$-0.418357\pi$
$61$	3.96271	0.507374	0.253687	+	0.967286i	$0.418357\pi$
$62$	6.37534i	0.809669i
$63$	− 8.64059i	− 1.08861i
$64$	−27.9567	−3.49458
$65$	0	0
$66$	27.5918	3.39632
$67$	6.80664i	0.831563i	0.909464	+	0.415782i	$0.136492\pi$
$67$	6.80664i	0.831563i	−0.909464	+	0.415782i	$0.863508\pi$
$68$	12.3905i	1.50257i
$69$	−0.608215	−0.0732205
$70$	0	0
$71$	−5.79485	−0.687722	−0.343861	−	0.939020i	$-0.611735\pi$
$71$	−5.79485	−0.687722	−0.343861	+	0.939020i	$0.611735\pi$
$72$	21.0186i	2.47706i
$73$	0.140654i	0.0164623i	0.999966	+	0.00823116i	$0.00262009\pi$
$73$	0.140654i	0.0164623i	−0.999966	+	0.00823116i	$0.997380\pi$
$74$	−2.72362	−0.316614
$75$	0	0
$76$	−12.9512	−1.48561
$77$	− 16.9085i	− 1.92690i
$78$	− 22.9563i	− 2.59928i
$79$	6.62418	0.745278	0.372639	−	0.927976i	$-0.378453\pi$
$79$	6.62418	0.745278	0.372639	+	0.927976i	$0.378453\pi$
$80$	0	0
$81$	−10.6758	−1.18620
$82$	− 12.0332i	− 1.32884i
$83$	− 13.9904i	− 1.53565i	−0.640660	−	0.767825i	$-0.721339\pi$
$83$	− 13.9904i	− 1.53565i	0.640660	−	0.767825i	$-0.278661\pi$
$84$	47.5462	5.18771
$85$	0	0
$86$	21.0059	2.26512
$87$	15.1014i	1.61904i
$88$	41.1305i	4.38453i
$89$	−14.8139	−1.57027	−0.785136	−	0.619323i	$-0.787407\pi$
$89$	−14.8139	−1.57027	−0.785136	+	0.619323i	$0.787407\pi$
$90$	0	0
$91$	−14.0678	−1.47470
$92$	− 1.43716i	− 0.149834i
$93$	5.36731i	0.556565i
$94$	−29.7651	−3.07004
$95$	0	0
$96$	−47.9839	−4.89734
$97$	− 8.94394i	− 0.908119i	−0.890971	−	0.454060i	$-0.849975\pi$
$97$	− 8.94394i	− 0.908119i	0.890971	−	0.454060i	$-0.150025\pi$
$98$	− 20.8266i	− 2.10381i
$99$	9.97490	1.00252
$100$	0	0
$101$	−11.5935	−1.15359	−0.576796	−	0.816888i	$-0.695697\pi$
$101$	−11.5935	−1.15359	−0.576796	+	0.816888i	$0.695697\pi$
$102$	14.2820i	1.41413i
$103$	13.5918i	1.33924i	0.742703	+	0.669620i	$0.233543\pi$
$103$	13.5918i	1.33924i	−0.742703	+	0.669620i	$0.766457\pi$
$104$	34.2204	3.35559
$105$	0	0
$106$	−0.298438	−0.0289869
$107$	6.66146i	0.643988i	0.946742	+	0.321994i	$0.104353\pi$
$107$	6.66146i	0.643988i	−0.946742	+	0.321994i	$0.895647\pi$
$108$	− 9.22151i	− 0.887340i
$109$	4.85120	0.464661	0.232331	−	0.972637i	$-0.425365\pi$
$109$	4.85120	0.464661	0.232331	+	0.972637i	$0.425365\pi$
$110$	0	0
$111$	−2.29298	−0.217640
$112$	55.5676i	5.25065i
$113$	2.83965i	0.267132i	0.991040	+	0.133566i	$0.0426428\pi$
$113$	2.83965i	0.267132i	−0.991040	+	0.133566i	$0.957357\pi$
$114$	−14.9283	−1.39816
$115$	0	0
$116$	−35.6833	−3.31311
$117$	− 8.29907i	− 0.767249i
$118$	5.46096i	0.502722i
$119$	8.75213	0.802306
$120$	0	0
$121$	8.51952	0.774502
$122$	10.7929i	0.977145i
$123$	− 10.1306i	− 0.913443i
$124$	−12.6825	−1.13892
$125$	0	0
$126$	23.5337	2.09655
$127$	8.81853i	0.782518i	0.920281	+	0.391259i	$0.127960\pi$
$127$	8.81853i	0.782518i	−0.920281	+	0.391259i	$0.872040\pi$
$128$	− 34.2903i	− 3.03087i
$129$	17.6845	1.55704
$130$	0	0
$131$	4.61549	0.403257	0.201629	−	0.979462i	$-0.435377\pi$
$131$	4.61549	0.403257	0.201629	+	0.979462i	$0.435377\pi$
$132$	54.8884i	4.77742i
$133$	9.14817i	0.793247i
$134$	−18.5387	−1.60150
$135$	0	0
$136$	−21.2899	−1.82559
$137$	− 14.1324i	− 1.20741i	−0.797207	−	0.603706i	$-0.793690\pi$
$137$	− 14.1324i	− 1.20741i	0.797207	−	0.603706i	$-0.206310\pi$
$138$	− 1.65655i	− 0.141015i
$139$	−13.2152	−1.12090	−0.560448	−	0.828190i	$-0.689371\pi$
$139$	−13.2152	−1.12090	−0.560448	+	0.828190i	$0.689371\pi$
$140$	0	0
$141$	−25.0589	−2.11034
$142$	− 15.7830i	− 1.32448i
$143$	− 16.2402i	− 1.35807i
$144$	−32.7813	−2.73178
$145$	0	0
$146$	−0.383088	−0.0317046
$147$	− 17.5337i	− 1.44615i
$148$	− 5.41809i	− 0.445364i
$149$	−6.99743	−0.573252	−0.286626	−	0.958043i	$-0.592534\pi$
$149$	−6.99743	−0.573252	−0.286626	+	0.958043i	$0.592534\pi$
$150$	0	0
$151$	19.7185	1.60467	0.802333	−	0.596877i	$-0.203592\pi$
$151$	19.7185	1.60467	0.802333	+	0.596877i	$0.203592\pi$
$152$	− 22.2533i	− 1.80498i
$153$	5.16319i	0.417419i
$154$	46.0522	3.71099
$155$	0	0
$156$	45.6669	3.65628
$157$	12.9768i	1.03566i	0.855482	+	0.517832i	$0.173261\pi$
$157$	12.9768i	1.03566i	−0.855482	+	0.517832i	$0.826739\pi$
$158$	18.0417i	1.43532i
$159$	−0.251251	−0.0199255
$160$	0	0
$161$	−1.01514	−0.0800045
$162$	− 29.0769i	− 2.28450i
$163$	2.33572i	0.182948i	0.995807	+	0.0914740i	$0.0291578\pi$
$163$	2.33572i	0.182948i	−0.995807	+	0.0914740i	$0.970842\pi$
$164$	23.9376	1.86921
$165$	0	0
$166$	38.1046	2.95749
$167$	− 22.8816i	− 1.77063i	−0.464994	−	0.885314i	$-0.653944\pi$
$167$	− 22.8816i	− 1.77063i	0.464994	−	0.885314i	$-0.346056\pi$
$168$	81.6956i	6.30295i
$169$	−0.511728	−0.0393637
$170$	0	0
$171$	−5.39682	−0.412705
$172$	41.7869i	3.18622i
$173$	− 14.0099i	− 1.06515i	−0.846382	−	0.532577i	$-0.821224\pi$
$173$	− 14.0099i	− 1.06515i	0.846382	−	0.532577i	$-0.178776\pi$
$174$	−41.1305	−3.11810
$175$	0	0
$176$	−64.1486	−4.83538
$177$	4.59751i	0.345570i
$178$	− 40.3474i	− 3.02417i
$179$	14.7815	1.10482	0.552412	−	0.833571i	$-0.313708\pi$
$179$	14.7815	1.10482	0.552412	+	0.833571i	$0.313708\pi$
$180$	0	0
$181$	−8.18949	−0.608720	−0.304360	−	0.952557i	$-0.598443\pi$
$181$	−8.18949	−0.608720	−0.304360	+	0.952557i	$0.598443\pi$
$182$	− 38.3152i	− 2.84011i
$183$	9.08641i	0.671687i
$184$	2.46938	0.182045
$185$	0	0
$186$	−14.6185	−1.07188
$187$	10.1037i	0.738853i
$188$	− 59.2118i	− 4.31846i
$189$	−6.51366	−0.473799
$190$	0	0
$191$	19.8743	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$191$	19.8743	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$192$	− 64.1040i	− 4.62631i
$193$	− 17.0390i	− 1.22650i	−0.789890	−	0.613249i	$-0.789862\pi$
$193$	− 17.0390i	− 1.22650i	0.789890	−	0.613249i	$-0.210138\pi$
$194$	24.3599	1.74894
$195$	0	0
$196$	41.4304	2.95932
$197$	− 26.5277i	− 1.89002i	−0.327038	−	0.945011i	$-0.606051\pi$
$197$	− 26.5277i	− 1.89002i	0.327038	−	0.945011i	$-0.393949\pi$
$198$	27.1678i	1.93073i
$199$	5.05687	0.358472	0.179236	−	0.983806i	$-0.442637\pi$
$199$	5.05687	0.358472	0.179236	+	0.983806i	$0.442637\pi$
$200$	0	0
$201$	−15.6075	−1.10087
$202$	− 31.5762i	− 2.22169i
$203$	25.2051i	1.76905i
$204$	−28.4112	−1.98918
$205$	0	0
$206$	−37.0189	−2.57923
$207$	− 0.598869i	− 0.0416243i
$208$	53.3713i	3.70063i
$209$	−10.5609	−0.730509
$210$	0	0
$211$	−19.6605	−1.35348	−0.676741	−	0.736221i	$-0.736608\pi$
$211$	−19.6605	−1.35348	−0.676741	+	0.736221i	$0.736608\pi$
$212$	− 0.593683i	− 0.0407743i
$213$	− 13.2875i	− 0.910442i
$214$	−18.1433	−1.24025
$215$	0	0
$216$	15.8447	1.07810
$217$	8.95833i	0.608131i
$218$	13.2128i	0.894885i
$219$	−0.322517	−0.0217937
$220$	0	0
$221$	8.40620	0.565462
$222$	− 6.24519i	− 0.419150i
$223$	24.1933i	1.62010i	0.586358	+	0.810052i	$0.300561\pi$
$223$	24.1933i	1.62010i	−0.586358	+	0.810052i	$0.699439\pi$
$224$	−80.0877	−5.35109
$225$	0	0
$226$	−7.73412	−0.514466
$227$	18.1013i	1.20143i	0.799465	+	0.600713i	$0.205117\pi$
$227$	18.1013i	1.20143i	−0.799465	+	0.600713i	$0.794883\pi$
$228$	− 29.6968i	− 1.96672i
$229$	−19.6182	−1.29641	−0.648205	−	0.761466i	$-0.724480\pi$
$229$	−19.6182	−1.29641	−0.648205	+	0.761466i	$0.724480\pi$
$230$	0	0
$231$	38.7707	2.55093
$232$	− 61.3124i	− 4.02536i
$233$	4.67013i	0.305951i	0.988230	+	0.152975i	$0.0488855\pi$
$233$	4.67013i	0.305951i	−0.988230	+	0.152975i	$0.951114\pi$
$234$	22.6035	1.47764
$235$	0	0
$236$	−10.8635	−0.707152
$237$	15.1891i	0.986637i
$238$	23.8375i	1.54515i
$239$	9.64645	0.623977	0.311989	−	0.950086i	$-0.399005\pi$
$239$	9.64645	0.623977	0.311989	+	0.950086i	$0.399005\pi$
$240$	0	0
$241$	1.59826	0.102953	0.0514764	−	0.998674i	$-0.483607\pi$
$241$	1.59826	0.102953	0.0514764	+	0.998674i	$0.483607\pi$
$242$	23.2039i	1.49160i
$243$	− 19.3735i	− 1.24281i
$244$	−21.4703	−1.37450
$245$	0	0
$246$	27.5918	1.75919
$247$	8.78658i	0.559077i
$248$	− 21.7915i	− 1.38376i
$249$	32.0797	2.03297
$250$	0	0
$251$	−6.49812	−0.410158	−0.205079	−	0.978745i	$-0.565745\pi$
$251$	−6.49812	−0.410158	−0.205079	+	0.978745i	$0.565745\pi$
$252$	46.8155i	2.94910i
$253$	− 1.17190i	− 0.0736770i
$254$	−24.0183	−1.50704
$255$	0	0
$256$	37.4804	2.34252
$257$	− 4.01155i	− 0.250234i	−0.992142	−	0.125117i	$-0.960069\pi$
$257$	− 4.01155i	− 0.250234i	0.992142	−	0.125117i	$-0.0399306\pi$
$258$	48.1659i	2.99868i
$259$	−3.82710	−0.237804
$260$	0	0
$261$	−14.8694	−0.920391
$262$	12.5708i	0.776629i
$263$	26.2537i	1.61887i	0.587208	+	0.809436i	$0.300227\pi$
$263$	26.2537i	1.61887i	−0.587208	+	0.809436i	$0.699773\pi$
$264$	−94.3113	−5.80446
$265$	0	0
$266$	−24.9161	−1.52770
$267$	− 33.9680i	− 2.07881i
$268$	− 36.8790i	− 2.25274i
$269$	−3.00335	−0.183117	−0.0915587	−	0.995800i	$-0.529185\pi$
$269$	−3.00335	−0.183117	−0.0915587	+	0.995800i	$0.529185\pi$
$270$	0	0
$271$	5.91480	0.359298	0.179649	−	0.983731i	$-0.442504\pi$
$271$	5.91480	0.359298	0.179649	+	0.983731i	$0.442504\pi$
$272$	− 33.2045i	− 2.01332i
$273$	− 32.2571i	− 1.95229i
$274$	38.4912	2.32534
$275$	0	0
$276$	3.29537	0.198358
$277$	12.2719i	0.737345i	0.929559	+	0.368672i	$0.120188\pi$
$277$	12.2719i	0.737345i	−0.929559	+	0.368672i	$0.879812\pi$
$278$	− 35.9931i	− 2.15872i
$279$	−5.28483	−0.316395
$280$	0	0
$281$	−3.42907	−0.204561	−0.102281	−	0.994756i	$-0.532614\pi$
$281$	−3.42907	−0.204561	−0.102281	+	0.994756i	$0.532614\pi$
$282$	− 68.2507i	− 4.06427i
$283$	21.8283i	1.29756i	0.760976	+	0.648780i	$0.224720\pi$
$283$	21.8283i	1.29756i	−0.760976	+	0.648780i	$0.775280\pi$
$284$	31.3970	1.86307
$285$	0	0
$286$	44.2320	2.61549
$287$	− 16.9085i	− 0.998075i
$288$	− 47.2466i	− 2.78403i
$289$	11.7702	0.692362
$290$	0	0
$291$	20.5082	1.20221
$292$	− 0.762077i	− 0.0445972i
$293$	7.43221i	0.434194i	0.976150	+	0.217097i	$0.0696589\pi$
$293$	7.43221i	0.434194i	−0.976150	+	0.217097i	$0.930341\pi$
$294$	47.7550	2.78513
$295$	0	0
$296$	9.30957	0.541108
$297$	− 7.51952i	− 0.436327i
$298$	− 19.0583i	− 1.10402i
$299$	−0.975019	−0.0563868
$300$	0	0
$301$	29.5164	1.70130
$302$	53.7055i	3.09041i
$303$	− 26.5835i	− 1.52718i
$304$	34.7070	1.99058
$305$	0	0
$306$	−14.0625	−0.803902
$307$	0.195164i	0.0111386i	0.999984	+	0.00556930i	$0.00177277\pi$
$307$	0.195164i	0.0111386i	−0.999984	+	0.00556930i	$0.998227\pi$
$308$	91.6116i	5.22006i
$309$	−31.1657	−1.77295
$310$	0	0
$311$	−9.98641	−0.566277	−0.283139	−	0.959079i	$-0.591376\pi$
$311$	−9.98641	−0.566277	−0.283139	+	0.959079i	$0.591376\pi$
$312$	78.4666i	4.44229i
$313$	− 5.33350i	− 0.301467i	−0.988574	−	0.150734i	$-0.951836\pi$
$313$	− 5.33350i	− 0.301467i	0.988574	−	0.150734i	$-0.0481635\pi$
$314$	−35.3439	−1.99457
$315$	0	0
$316$	−35.8904	−2.01899
$317$	24.8111i	1.39353i	0.717300	+	0.696765i	$0.245378\pi$
$317$	24.8111i	1.39353i	−0.717300	+	0.696765i	$0.754622\pi$
$318$	− 0.684312i	− 0.0383743i
$319$	−29.0973	−1.62914
$320$	0	0
$321$	−15.2746	−0.852544
$322$	− 2.76486i	− 0.154080i
$323$	− 5.46649i	− 0.304164i
$324$	57.8426	3.21348
$325$	0	0
$326$	−6.36162	−0.352337
$327$	11.1237i	0.615142i
$328$	41.1305i	2.27105i
$329$	−41.8246	−2.30586
$330$	0	0
$331$	−15.8959	−0.873717	−0.436859	−	0.899530i	$-0.643909\pi$
$331$	−15.8959	−0.873717	−0.436859	+	0.899530i	$0.643909\pi$
$332$	75.8015i	4.16015i
$333$	− 2.25774i	− 0.123723i
$334$	62.3206	3.41003
$335$	0	0
$336$	−127.415	−6.95107
$337$	25.2470i	1.37529i	0.726045	+	0.687647i	$0.241356\pi$
$337$	25.2470i	1.37529i	−0.726045	+	0.687647i	$0.758644\pi$
$338$	− 1.39375i	− 0.0758101i
$339$	−6.51125	−0.353642
$340$	0	0
$341$	−10.3417	−0.560035
$342$	− 14.6989i	− 0.794824i
$343$	− 2.47493i	− 0.133634i
$344$	−71.7999	−3.87119
$345$	0	0
$346$	38.1576	2.05137
$347$	23.4864i	1.26082i	0.776264	+	0.630408i	$0.217112\pi$
$347$	23.4864i	1.26082i	−0.776264	+	0.630408i	$0.782888\pi$
$348$	− 81.8209i	− 4.38606i
$349$	7.13314	0.381828	0.190914	−	0.981607i	$-0.438855\pi$
$349$	7.13314	0.381828	0.190914	+	0.981607i	$0.438855\pi$
$350$	0	0
$351$	−6.25621	−0.333932
$352$	− 92.4551i	− 4.92787i
$353$	1.49954i	0.0798126i	0.999203	+	0.0399063i	$0.0127059\pi$
$353$	1.49954i	0.0798126i	−0.999203	+	0.0399063i	$0.987294\pi$
$354$	−12.5218	−0.665529
$355$	0	0
$356$	80.2631	4.25394
$357$	20.0684i	1.06213i
$358$	40.2593i	2.12777i
$359$	6.47638	0.341810	0.170905	−	0.985287i	$-0.445331\pi$
$359$	6.47638	0.341810	0.170905	+	0.985287i	$0.445331\pi$
$360$	0	0
$361$	−13.2862	−0.699271
$362$	− 22.3050i	− 1.17233i
$363$	19.5351i	1.02532i
$364$	76.2204	3.99504
$365$	0	0
$366$	−24.7479	−1.29359
$367$	24.5122i	1.27953i	0.768572	+	0.639763i	$0.220967\pi$
$367$	24.5122i	1.27953i	−0.768572	+	0.639763i	$0.779033\pi$
$368$	3.85132i	0.200764i
$369$	9.97490	0.519272
$370$	0	0
$371$	−0.419351	−0.0217716
$372$	− 29.0806i	− 1.50776i
$373$	18.9696i	0.982208i	0.871101	+	0.491104i	$0.163407\pi$
$373$	18.9696i	0.982208i	−0.871101	+	0.491104i	$0.836593\pi$
$374$	−27.5185	−1.42295
$375$	0	0
$376$	101.740	5.24684
$377$	24.2088i	1.24682i
$378$	− 17.7407i	− 0.912485i
$379$	−23.7286	−1.21886	−0.609428	−	0.792841i	$-0.708601\pi$
$379$	−23.7286	−1.21886	−0.609428	+	0.792841i	$0.708601\pi$
$380$	0	0
$381$	−20.2207	−1.03594
$382$	54.1299i	2.76953i
$383$	12.0392i	0.615173i	0.951520	+	0.307586i	$0.0995213\pi$
$383$	12.0392i	0.615173i	−0.951520	+	0.307586i	$0.900479\pi$
$384$	78.6269	4.01241
$385$	0	0
$386$	46.4078	2.36210
$387$	17.4128i	0.885142i
$388$	48.4591i	2.46014i
$389$	−30.3044	−1.53649	−0.768247	−	0.640153i	$-0.778871\pi$
$389$	−30.3044	−1.53649	−0.768247	+	0.640153i	$0.778871\pi$
$390$	0	0
$391$	0.606599	0.0306770
$392$	71.1874i	3.59550i
$393$	10.5832i	0.533853i
$394$	72.2514	3.63997
$395$	0	0
$396$	−54.0449	−2.71586
$397$	− 26.5246i	− 1.33123i	−0.746295	−	0.665615i	$-0.768169\pi$
$397$	− 26.5246i	− 1.33123i	0.746295	−	0.665615i	$-0.231831\pi$
$398$	13.7730i	0.690377i
$399$	−20.9765	−1.05014
$400$	0	0
$401$	15.7889	0.788462	0.394231	−	0.919011i	$-0.371011\pi$
$401$	15.7889	0.788462	0.394231	+	0.919011i	$0.371011\pi$
$402$	− 42.5088i	− 2.12014i
$403$	8.60425i	0.428608i
$404$	62.8144	3.12513
$405$	0	0
$406$	−68.6490	−3.40699
$407$	− 4.41809i	− 0.218997i
$408$	− 48.8172i	− 2.41681i
$409$	9.75340	0.482275	0.241137	−	0.970491i	$-0.422480\pi$
$409$	9.75340	0.482275	0.241137	+	0.970491i	$0.422480\pi$
$410$	0	0
$411$	32.4052	1.59843
$412$	− 73.6417i	− 3.62806i
$413$	7.67348i	0.377587i
$414$	1.63109	0.0801637
$415$	0	0
$416$	−76.9222	−3.77142
$417$	− 30.3021i	− 1.48390i
$418$	− 28.7637i	− 1.40688i
$419$	9.64020	0.470954	0.235477	−	0.971880i	$-0.424335\pi$
$419$	9.64020	0.470954	0.235477	+	0.971880i	$0.424335\pi$
$420$	0	0
$421$	−17.5464	−0.855161	−0.427581	−	0.903977i	$-0.640634\pi$
$421$	−17.5464	−0.855161	−0.427581	+	0.903977i	$0.640634\pi$
$422$	− 53.5476i	− 2.60666i
$423$	− 24.6738i	− 1.19968i
$424$	1.02009	0.0495399
$425$	0	0
$426$	36.1899	1.75341
$427$	15.1657i	0.733920i
$428$	− 36.0924i	− 1.74459i
$429$	37.2383	1.79788
$430$	0	0
$431$	2.48113	0.119512	0.0597558	−	0.998213i	$-0.480968\pi$
$431$	2.48113	0.119512	0.0597558	+	0.998213i	$0.480968\pi$
$432$	24.7120i	1.18896i
$433$	18.3658i	0.882603i	0.897359	+	0.441302i	$0.145483\pi$
$433$	18.3658i	0.882603i	−0.897359	+	0.441302i	$0.854517\pi$
$434$	−24.3991	−1.17119
$435$	0	0
$436$	−26.2843	−1.25879
$437$	0.634048i	0.0303306i
$438$	− 0.878412i	− 0.0419721i
$439$	−23.5970	−1.12622	−0.563111	−	0.826381i	$-0.690396\pi$
$439$	−23.5970	−1.12622	−0.563111	+	0.826381i	$0.690396\pi$
$440$	0	0
$441$	17.2642	0.822106
$442$	22.8953i	1.08902i
$443$	1.45360i	0.0690625i	0.999404	+	0.0345313i	$0.0109938\pi$
$443$	1.45360i	0.0690625i	−0.999404	+	0.0345313i	$0.989006\pi$
$444$	12.4236	0.589596
$445$	0	0
$446$	−65.8933	−3.12014
$447$	− 16.0449i	− 0.758900i
$448$	− 106.993i	− 5.05494i
$449$	38.3957	1.81200	0.906001	−	0.423275i	$-0.139119\pi$
$449$	38.3957	1.81200	0.906001	+	0.423275i	$0.139119\pi$
$450$	0	0
$451$	19.5195	0.919138
$452$	− 15.3855i	− 0.723672i
$453$	45.2140i	2.12434i
$454$	−49.3010	−2.31381
$455$	0	0
$456$	51.0262	2.38952
$457$	20.1534i	0.942736i	0.881937	+	0.471368i	$0.156240\pi$
$457$	20.1534i	0.942736i	−0.881937	+	0.471368i	$0.843760\pi$
$458$	− 53.4326i	− 2.49674i
$459$	3.89224	0.181674
$460$	0	0
$461$	34.6000	1.61148	0.805742	−	0.592267i	$-0.201767\pi$
$461$	34.6000	1.61148	0.805742	+	0.592267i	$0.201767\pi$
$462$	105.597i	4.91280i
$463$	− 33.7416i	− 1.56810i	−0.620695	−	0.784052i	$-0.713150\pi$
$463$	− 33.7416i	− 1.56810i	0.620695	−	0.784052i	$-0.286850\pi$
$464$	95.6249	4.43927
$465$	0	0
$466$	−12.7197	−0.589227
$467$	− 32.4863i	− 1.50329i	−0.659570	−	0.751643i	$-0.729262\pi$
$467$	− 32.4863i	− 1.50329i	0.659570	−	0.751643i	$-0.270738\pi$
$468$	44.9651i	2.07851i
$469$	−26.0497	−1.20286
$470$	0	0
$471$	−29.7556	−1.37106
$472$	− 18.6661i	− 0.859175i
$473$	34.0745i	1.56675i
$474$	−41.3692	−1.90015
$475$	0	0
$476$	−47.4198	−2.17348
$477$	− 0.247390i	− 0.0113272i
$478$	26.2732i	1.20171i
$479$	−19.3799	−0.885492	−0.442746	−	0.896647i	$-0.645996\pi$
$479$	−19.3799	−0.885492	−0.442746	+	0.896647i	$0.645996\pi$
$480$	0	0
$481$	−3.67583	−0.167603
$482$	4.35304i	0.198276i
$483$	− 2.32770i	− 0.105914i
$484$	−46.1595	−2.09816
$485$	0	0
$486$	52.7659	2.39351
$487$	2.36162i	0.107015i	0.998567	+	0.0535076i	$0.0170401\pi$
$487$	2.36162i	0.107015i	−0.998567	+	0.0535076i	$0.982960\pi$
$488$	− 36.8912i	− 1.66998i
$489$	−5.35576	−0.242196
$490$	0	0
$491$	14.1469	0.638442	0.319221	−	0.947680i	$-0.396579\pi$
$491$	14.1469	0.638442	0.319221	+	0.947680i	$0.396579\pi$
$492$	54.8884i	2.47456i
$493$	− 15.0613i	− 0.678327i
$494$	−23.9313	−1.07672
$495$	0	0
$496$	33.9868	1.52605
$497$	− 22.1775i	− 0.994796i
$498$	87.3729i	3.91528i
$499$	31.1547	1.39468	0.697338	−	0.716743i	$-0.254368\pi$
$499$	31.1547	1.39468	0.697338	+	0.716743i	$0.254368\pi$
$500$	0	0
$501$	52.4669	2.34405
$502$	− 17.6984i	− 0.789918i
$503$	13.3094i	0.593438i	0.954965	+	0.296719i	$0.0958925\pi$
$503$	13.3094i	0.593438i	−0.954965	+	0.296719i	$0.904108\pi$
$504$	−80.4402	−3.58309
$505$	0	0
$506$	3.19182	0.141894
$507$	− 1.17338i	− 0.0521117i
$508$	− 47.7796i	− 2.11988i
$509$	−24.4420	−1.08337	−0.541686	−	0.840581i	$-0.682214\pi$
$509$	−24.4420	−1.08337	−0.541686	+	0.840581i	$0.682214\pi$
$510$	0	0
$511$	−0.538297	−0.0238129
$512$	33.5015i	1.48057i
$513$	4.06837i	0.179623i
$514$	10.9259	0.481922
$515$	0	0
$516$	−95.8165	−4.21808
$517$	− 48.2832i	− 2.12349i
$518$	− 10.4236i	− 0.457985i
$519$	32.1244	1.41010
$520$	0	0
$521$	21.2938	0.932897	0.466448	−	0.884548i	$-0.345533\pi$
$521$	21.2938	0.932897	0.466448	+	0.884548i	$0.345533\pi$
$522$	− 40.4985i	− 1.77257i
$523$	4.62687i	0.202319i	0.994870	+	0.101159i	$0.0322552\pi$
$523$	4.62687i	0.202319i	−0.994870	+	0.101159i	$0.967745\pi$
$524$	−25.0072	−1.09244
$525$	0	0
$526$	−71.5050	−3.11777
$527$	− 5.35305i	− 0.233183i
$528$	− 147.091i	− 6.40132i
$529$	22.9296	0.996941
$530$	0	0
$531$	−4.52686	−0.196449
$532$	− 49.5656i	− 2.14894i
$533$	− 16.2402i	− 0.703439i
$534$	92.5157	4.00355
$535$	0	0
$536$	63.3669	2.73703
$537$	33.8937i	1.46262i
$538$	− 8.17997i	− 0.352664i
$539$	33.7838	1.45517
$540$	0	0
$541$	−18.9838	−0.816179	−0.408089	−	0.912942i	$-0.633805\pi$
$541$	−18.9838	−0.816179	−0.408089	+	0.912942i	$0.633805\pi$
$542$	16.1096i	0.691968i
$543$	− 18.7783i	− 0.805855i
$544$	47.8564	2.05183
$545$	0	0
$546$	87.8559	3.75988
$547$	3.76479i	0.160971i	0.996756	+	0.0804855i	$0.0256471\pi$
$547$	3.76479i	0.160971i	−0.996756	+	0.0804855i	$0.974353\pi$
$548$	76.5706i	3.27093i
$549$	−8.94678	−0.381839
$550$	0	0
$551$	15.7428	0.670667
$552$	5.66222i	0.241000i
$553$	25.3514i	1.07805i
$554$	−33.4239	−1.42004
$555$	0	0
$556$	71.6010	3.03656
$557$	− 29.2177i	− 1.23799i	−0.785394	−	0.618996i	$-0.787540\pi$
$557$	− 29.2177i	− 1.23799i	0.785394	−	0.618996i	$-0.212460\pi$
$558$	− 14.3939i	− 0.609341i
$559$	28.3498	1.19907
$560$	0	0
$561$	−23.1675	−0.978131
$562$	− 9.33948i	− 0.393962i
$563$	17.2201i	0.725741i	0.931840	+	0.362871i	$0.118203\pi$
$563$	17.2201i	0.725741i	−0.931840	+	0.362871i	$0.881797\pi$
$564$	135.771	5.71700
$565$	0	0
$566$	−59.4520	−2.49895
$567$	− 40.8575i	− 1.71585i
$568$	53.9476i	2.26359i
$569$	28.1495	1.18009	0.590045	−	0.807371i	$-0.299110\pi$
$569$	28.1495	1.18009	0.590045	+	0.807371i	$0.299110\pi$
$570$	0	0
$571$	29.0284	1.21480	0.607401	−	0.794396i	$-0.292212\pi$
$571$	29.0284	1.21480	0.607401	+	0.794396i	$0.292212\pi$
$572$	87.9906i	3.67907i
$573$	45.5713i	1.90377i
$574$	46.0522	1.92218
$575$	0	0
$576$	63.1189	2.62995
$577$	18.3603i	0.764347i	0.924090	+	0.382174i	$0.124824\pi$
$577$	18.3603i	0.764347i	−0.924090	+	0.382174i	$0.875176\pi$
$578$	32.0574i	1.33341i
$579$	39.0701	1.62370
$580$	0	0
$581$	53.5428	2.22133
$582$	55.8566i	2.31533i
$583$	− 0.484109i	− 0.0200497i
$584$	1.30943	0.0541846
$585$	0	0
$586$	−20.2425	−0.836210
$587$	− 40.9523i	− 1.69028i	−0.534544	−	0.845141i	$-0.679516\pi$
$587$	− 40.9523i	− 1.69028i	0.534544	−	0.845141i	$-0.320484\pi$
$588$	94.9990i	3.91769i
$589$	5.59528	0.230549
$590$	0	0
$591$	60.8275	2.50211
$592$	14.5195i	0.596749i
$593$	− 12.6520i	− 0.519557i	−0.965668	−	0.259778i	$-0.916350\pi$
$593$	− 12.6520i	− 0.519557i	0.965668	−	0.259778i	$-0.0836495\pi$
$594$	20.4803	0.840317
$595$	0	0
$596$	37.9127	1.55297
$597$	11.5953i	0.474563i
$598$	− 2.65558i	− 0.108595i
$599$	22.0493	0.900910	0.450455	−	0.892799i	$-0.351262\pi$
$599$	22.0493	0.900910	0.450455	+	0.892799i	$0.351262\pi$
$600$	0	0
$601$	0.498980	0.0203538	0.0101769	−	0.999948i	$-0.496761\pi$
$601$	0.498980	0.0203538	0.0101769	+	0.999948i	$0.496761\pi$
$602$	80.3915i	3.27651i
$603$	− 15.3676i	− 0.625818i
$604$	−106.836	−4.34711
$605$	0	0
$606$	72.4034	2.94119
$607$	41.2564i	1.67455i	0.546784	+	0.837274i	$0.315852\pi$
$607$	41.2564i	1.67455i	−0.546784	+	0.837274i	$0.684148\pi$
$608$	50.0219i	2.02866i
$609$	−57.7947	−2.34196
$610$	0	0
$611$	−40.1714	−1.62516
$612$	− 27.9746i	− 1.13081i
$613$	4.13184i	0.166883i	0.996513	+	0.0834417i	$0.0265912\pi$
$613$	4.13184i	0.166883i	−0.996513	+	0.0834417i	$0.973409\pi$
$614$	−0.531552	−0.0214517
$615$	0	0
$616$	−157.411	−6.34225
$617$	37.7095i	1.51813i	0.651017	+	0.759063i	$0.274343\pi$
$617$	37.7095i	1.51813i	−0.651017	+	0.759063i	$0.725657\pi$
$618$	− 84.8834i	− 3.41451i
$619$	23.7226	0.953493	0.476746	−	0.879041i	$-0.341816\pi$
$619$	23.7226	0.953493	0.476746	+	0.879041i	$0.341816\pi$
$620$	0	0
$621$	−0.451454	−0.0181162
$622$	− 27.1992i	− 1.09059i
$623$	− 56.6943i	− 2.27141i
$624$	−122.379	−4.89909
$625$	0	0
$626$	14.5264	0.580592
$627$	− 24.2158i	− 0.967085i
$628$	− 70.3097i	− 2.80566i
$629$	2.28688	0.0911840
$630$	0	0
$631$	21.4175	0.852615	0.426308	−	0.904578i	$-0.359814\pi$
$631$	21.4175	0.852615	0.426308	+	0.904578i	$0.359814\pi$
$632$	− 61.6682i	− 2.45303i
$633$	− 45.0810i	− 1.79181i
$634$	−67.5759	−2.68378
$635$	0	0
$636$	1.36130	0.0539791
$637$	− 28.1079i	− 1.11368i
$638$	− 79.2500i	− 3.13754i
$639$	13.0833	0.517566
$640$	0	0
$641$	−36.6406	−1.44722	−0.723608	−	0.690211i	$-0.757518\pi$
$641$	−36.6406	−1.44722	−0.723608	+	0.690211i	$0.757518\pi$
$642$	− 41.6021i	− 1.64190i
$643$	− 13.5452i	− 0.534170i	−0.963673	−	0.267085i	$-0.913940\pi$
$643$	− 13.5452i	− 0.534170i	0.963673	−	0.267085i	$-0.0860604\pi$
$644$	5.50014	0.216736
$645$	0	0
$646$	14.8886	0.585785
$647$	18.8904i	0.742657i	0.928502	+	0.371329i	$0.121098\pi$
$647$	18.8904i	0.742657i	−0.928502	+	0.371329i	$0.878902\pi$
$648$	99.3874i	3.90431i
$649$	−8.85844	−0.347724
$650$	0	0
$651$	−20.5412	−0.805075
$652$	− 12.6552i	− 0.495614i
$653$	7.44489i	0.291341i	0.989333	+	0.145670i	$0.0465339\pi$
$653$	7.44489i	0.291341i	−0.989333	+	0.145670i	$0.953466\pi$
$654$	−30.2967	−1.18469
$655$	0	0
$656$	−64.1486	−2.50458
$657$	− 0.317560i	− 0.0123892i
$658$	− 113.914i	− 4.44083i
$659$	−5.00068	−0.194799	−0.0973993	−	0.995245i	$-0.531052\pi$
$659$	−5.00068	−0.194799	−0.0973993	+	0.995245i	$0.531052\pi$
$660$	0	0
$661$	−7.47493	−0.290741	−0.145371	−	0.989377i	$-0.546437\pi$
$661$	−7.47493	−0.290741	−0.145371	+	0.989377i	$0.546437\pi$
$662$	− 43.2943i	− 1.68268i
$663$	19.2752i	0.748587i
$664$	−130.245	−5.05448
$665$	0	0
$666$	6.14922	0.238277
$667$	1.74693i	0.0676415i
$668$	123.974i	4.79671i
$669$	−55.4747	−2.14477
$670$	0	0
$671$	−17.5076	−0.675875
$672$	− 183.639i	− 7.08404i
$673$	38.6412i	1.48951i	0.667338	+	0.744755i	$0.267434\pi$
$673$	38.6412i	1.48951i	−0.667338	+	0.744755i	$0.732566\pi$
$674$	−68.7632	−2.64866
$675$	0	0
$676$	2.77259	0.106638
$677$	43.8791i	1.68641i	0.537590	+	0.843206i	$0.319335\pi$
$677$	43.8791i	1.68641i	−0.537590	+	0.843206i	$0.680665\pi$
$678$	− 17.7341i	− 0.681076i
$679$	34.2293	1.31360
$680$	0	0
$681$	−41.5059	−1.59051
$682$	− 28.1668i	− 1.07856i
$683$	− 18.8393i	− 0.720867i	−0.932785	−	0.360433i	$-0.882629\pi$
$683$	− 18.8393i	− 0.720867i	0.932785	−	0.360433i	$-0.117371\pi$
$684$	29.2405	1.11804
$685$	0	0
$686$	6.74077	0.257364
$687$	− 44.9842i	− 1.71625i
$688$	− 111.982i	− 4.26926i
$689$	−0.402776	−0.0153445
$690$	0	0
$691$	−42.3770	−1.61210	−0.806048	−	0.591850i	$-0.798398\pi$
$691$	−42.3770	−1.61210	−0.806048	+	0.591850i	$0.798398\pi$
$692$	75.9069i	2.88555i
$693$	38.1749i	1.45015i
$694$	−63.9680	−2.42819
$695$	0	0
$696$	140.588	5.32897
$697$	10.1037i	0.382703i
$698$	19.4279i	0.735359i
$699$	−10.7085	−0.405033
$700$	0	0
$701$	16.1386	0.609546	0.304773	−	0.952425i	$-0.401419\pi$
$701$	16.1386	0.609546	0.304773	+	0.952425i	$0.401419\pi$
$702$	− 17.0395i	− 0.643115i
$703$	2.39037i	0.0901543i
$704$	123.515	4.65515
$705$	0	0
$706$	−4.08418	−0.153710
$707$	− 44.3693i	− 1.66868i
$708$	− 24.9097i	− 0.936164i
$709$	20.5619	0.772218	0.386109	−	0.922453i	$-0.373819\pi$
$709$	20.5619	0.772218	0.386109	+	0.922453i	$0.373819\pi$
$710$	0	0
$711$	−14.9557	−0.560881
$712$	137.911i	5.16844i
$713$	0.620891i	0.0232525i
$714$	−54.6587	−2.04555
$715$	0	0
$716$	−80.0877	−2.99302
$717$	22.1191i	0.826052i
$718$	17.6392i	0.658288i
$719$	−5.93966	−0.221512	−0.110756	−	0.993848i	$-0.535327\pi$
$719$	−5.93966	−0.221512	−0.110756	+	0.993848i	$0.535327\pi$
$720$	0	0
$721$	−52.0172	−1.93722
$722$	− 36.1864i	− 1.34672i
$723$	3.66477i	0.136294i
$724$	44.3714	1.64905
$725$	0	0
$726$	−53.2060	−1.97466
$727$	− 24.8976i	− 0.923401i	−0.887036	−	0.461701i	$-0.847239\pi$
$727$	− 24.8976i	− 0.923401i	0.887036	−	0.461701i	$-0.152761\pi$
$728$	130.965i	4.85388i
$729$	12.3954	0.459089
$730$	0	0
$731$	−17.6376	−0.652349
$732$	− 49.2310i	− 1.81963i
$733$	− 20.8362i	− 0.769604i	−0.922999	−	0.384802i	$-0.874270\pi$
$733$	− 20.8362i	− 0.769604i	0.922999	−	0.384802i	$-0.125730\pi$
$734$	−66.7618	−2.46422
$735$	0	0
$736$	−5.55078	−0.204604
$737$	− 30.0724i	− 1.10773i
$738$	27.1678i	1.00006i
$739$	39.3500	1.44751	0.723756	−	0.690056i	$-0.242414\pi$
$739$	39.3500	1.44751	0.723756	+	0.690056i	$0.242414\pi$
$740$	0	0
$741$	−20.1474	−0.740134
$742$	− 1.14215i	− 0.0419297i
$743$	− 3.25437i	− 0.119391i	−0.998217	−	0.0596955i	$-0.980987\pi$
$743$	− 3.25437i	− 0.119391i	0.998217	−	0.0596955i	$-0.0190130\pi$
$744$	49.9674	1.83189
$745$	0	0
$746$	−51.6659	−1.89162
$747$	31.5868i	1.15570i
$748$	− 54.7425i	− 2.00159i
$749$	−25.4941	−0.931533
$750$	0	0
$751$	45.4895	1.65994	0.829968	−	0.557811i	$-0.188358\pi$
$751$	45.4895	1.65994	0.829968	+	0.557811i	$0.188358\pi$
$752$	158.677i	5.78636i
$753$	− 14.9000i	− 0.542988i
$754$	−65.9356	−2.40123
$755$	0	0
$756$	35.2916	1.28354
$757$	31.5740i	1.14758i	0.819004	+	0.573788i	$0.194527\pi$
$757$	31.5740i	1.14758i	−0.819004	+	0.573788i	$0.805473\pi$
$758$	− 64.6276i	− 2.34738i
$759$	2.68715	0.0975374
$760$	0	0
$761$	41.8552	1.51725	0.758626	−	0.651527i	$-0.225871\pi$
$761$	41.8552	1.51725	0.758626	+	0.651527i	$0.225871\pi$
$762$	− 55.0734i	− 1.99510i
$763$	18.5660i	0.672136i
$764$	−107.681	−3.89575
$765$	0	0
$766$	−32.7901	−1.18475
$767$	7.37018i	0.266122i
$768$	85.9416i	3.10115i
$769$	22.1619	0.799180	0.399590	−	0.916694i	$-0.369152\pi$
$769$	22.1619	0.799180	0.399590	+	0.916694i	$0.369152\pi$
$770$	0	0
$771$	9.19840	0.331272
$772$	92.3191i	3.32264i
$773$	− 19.3276i	− 0.695165i	−0.937649	−	0.347582i	$-0.887003\pi$
$773$	− 19.3276i	− 0.695165i	0.937649	−	0.347582i	$-0.112997\pi$
$774$	−47.4258	−1.70468
$775$	0	0
$776$	−83.2642	−2.98901
$777$	− 8.77545i	− 0.314817i
$778$	− 82.5376i	− 2.95912i
$779$	−10.5609	−0.378382
$780$	0	0
$781$	25.6022	0.916118
$782$	1.65214i	0.0590806i
$783$	11.2092i	0.400584i
$784$	−111.026	−3.96522
$785$	0	0
$786$	−28.8246	−1.02814
$787$	19.5111i	0.695496i	0.937588	+	0.347748i	$0.113054\pi$
$787$	19.5111i	0.695496i	−0.937588	+	0.347748i	$0.886946\pi$
$788$	143.730i	5.12016i
$789$	−60.1991	−2.14314
$790$	0	0
$791$	−10.8676	−0.386408
$792$	− 92.8620i	− 3.29971i
$793$	14.5663i	0.517263i
$794$	72.2428	2.56380
$795$	0	0
$796$	−27.3986	−0.971117
$797$	− 22.2367i	− 0.787664i	−0.919183	−	0.393832i	$-0.871149\pi$
$797$	− 22.2367i	− 0.787664i	0.919183	−	0.393832i	$-0.128851\pi$
$798$	− 57.1320i	− 2.02245i
$799$	24.9923	0.884163
$800$	0	0
$801$	33.4460	1.18176
$802$	43.0030i	1.51849i
$803$	− 0.621423i	− 0.0219295i
$804$	84.5627	2.98229
$805$	0	0
$806$	−23.4347	−0.825451
$807$	− 6.88661i	− 0.242420i
$808$	107.930i	3.79697i
$809$	−12.4896	−0.439110	−0.219555	−	0.975600i	$-0.570461\pi$
$809$	−12.4896	−0.439110	−0.219555	+	0.975600i	$0.570461\pi$
$810$	0	0
$811$	−3.99535	−0.140296	−0.0701478	−	0.997537i	$-0.522347\pi$
$811$	−3.99535	−0.140296	−0.0701478	+	0.997537i	$0.522347\pi$
$812$	− 136.563i	− 4.79244i
$813$	13.5625i	0.475657i
$814$	12.0332	0.421763
$815$	0	0
$816$	76.1370	2.66533
$817$	− 18.4357i	− 0.644982i
$818$	26.5645i	0.928807i
$819$	31.7614	1.10983
$820$	0	0
$821$	−1.51952	−0.0530317	−0.0265159	−	0.999648i	$-0.508441\pi$
$821$	−1.51952	−0.0530317	−0.0265159	+	0.999648i	$0.508441\pi$
$822$	88.2595i	3.07840i
$823$	− 46.9850i	− 1.63779i	−0.573940	−	0.818897i	$-0.694586\pi$
$823$	− 46.9850i	− 1.63779i	0.573940	−	0.818897i	$-0.305414\pi$
$824$	126.534	4.40802
$825$	0	0
$826$	−20.8996	−0.727191
$827$	− 10.5199i	− 0.365814i	−0.983130	−	0.182907i	$-0.941449\pi$
$827$	− 10.5199i	− 0.365814i	0.983130	−	0.182907i	$-0.0585507\pi$
$828$	3.24473i	0.112762i
$829$	53.8789	1.87129	0.935645	−	0.352942i	$-0.114819\pi$
$829$	53.8789	1.87129	0.935645	+	0.352942i	$0.114819\pi$
$830$	0	0
$831$	−28.1391	−0.976134
$832$	− 102.764i	− 3.56270i
$833$	17.4871i	0.605892i
$834$	82.5312	2.85782
$835$	0	0
$836$	57.2196	1.97898
$837$	3.98394i	0.137705i
$838$	26.2562i	0.907006i
$839$	−12.0120	−0.414699	−0.207349	−	0.978267i	$-0.566484\pi$
$839$	−12.0120	−0.414699	−0.207349	+	0.978267i	$0.566484\pi$
$840$	0	0
$841$	14.3748	0.495682
$842$	− 47.7898i	− 1.64694i
$843$	− 7.86278i	− 0.270808i
$844$	106.522	3.66665
$845$	0	0
$846$	67.2019	2.31045
$847$	32.6051i	1.12032i
$848$	1.59096i	0.0546340i
$849$	−50.0518	−1.71777
$850$	0	0
$851$	−0.265251	−0.00909270
$852$	71.9926i	2.46643i
$853$	0.967727i	0.0331343i	0.999863	+	0.0165672i	$0.00527373\pi$
$853$	0.967727i	0.0331343i	−0.999863	+	0.0165672i	$0.994726\pi$
$854$	−41.3056	−1.41345
$855$	0	0
$856$	62.0154	2.11964
$857$	0.683897i	0.0233615i	0.999932	+	0.0116807i	$0.00371818\pi$
$857$	0.683897i	0.0233615i	−0.999932	+	0.0116807i	$0.996282\pi$
$858$	101.423i	3.46252i
$859$	−23.9043	−0.815605	−0.407803	−	0.913070i	$-0.633705\pi$
$859$	−23.9043	−0.815605	−0.407803	+	0.913070i	$0.633705\pi$
$860$	0	0
$861$	38.7707	1.32130
$862$	6.75764i	0.230166i
$863$	− 51.4456i	− 1.75123i	−0.483012	−	0.875614i	$-0.660457\pi$
$863$	− 51.4456i	− 1.75123i	0.483012	−	0.875614i	$-0.339543\pi$
$864$	−35.6166	−1.21170
$865$	0	0
$866$	−50.0214	−1.69979
$867$	26.9887i	0.916584i
$868$	− 48.5371i	− 1.64745i
$869$	−29.2662	−0.992788
$870$	0	0
$871$	−25.0201	−0.847772
$872$	− 45.1626i	− 1.52940i
$873$	20.1931i	0.683433i
$874$	−1.72690	−0.0584134
$875$	0	0
$876$	1.74742	0.0590400
$877$	− 29.8721i	− 1.00871i	−0.863497	−	0.504354i	$-0.831731\pi$
$877$	− 29.8721i	− 1.00871i	0.863497	−	0.504354i	$-0.168269\pi$
$878$	− 64.2691i	− 2.16898i
$879$	−17.0419	−0.574808
$880$	0	0
$881$	25.1190	0.846279	0.423140	−	0.906064i	$-0.360928\pi$
$881$	25.1190	0.846279	0.423140	+	0.906064i	$0.360928\pi$
$882$	47.0211i	1.58328i
$883$	− 39.3961i	− 1.32578i	−0.748715	−	0.662892i	$-0.769329\pi$
$883$	− 39.3961i	− 1.32578i	0.748715	−	0.662892i	$-0.230671\pi$
$884$	−45.5455	−1.53186
$885$	0	0
$886$	−3.95904	−0.133007
$887$	18.7079i	0.628148i	0.949398	+	0.314074i	$0.101694\pi$
$887$	18.7079i	0.628148i	−0.949398	+	0.314074i	$0.898306\pi$
$888$	21.3466i	0.716346i
$889$	−33.7494	−1.13192
$890$	0	0
$891$	47.1668	1.58015
$892$	− 131.082i	− 4.38893i
$893$	26.1232i	0.874179i
$894$	43.7003	1.46156
$895$	0	0
$896$	131.232	4.38417
$897$	− 2.23570i	− 0.0746477i
$898$	104.575i	3.48971i
$899$	15.4162	0.514158
$900$	0	0
$901$	0.250583	0.00834815
$902$	53.1637i	1.77016i
$903$	67.6805i	2.25227i
$904$	26.4359	0.879245
$905$	0	0
$906$	−123.146	−4.09124
$907$	16.8702i	0.560164i	0.959976	+	0.280082i	$0.0903617\pi$
$907$	16.8702i	0.560164i	−0.959976	+	0.280082i	$0.909638\pi$
$908$	− 98.0745i	− 3.25472i
$909$	26.1750	0.868171
$910$	0	0
$911$	32.4128	1.07388	0.536941	−	0.843620i	$-0.319580\pi$
$911$	32.4128	1.07388	0.536941	+	0.843620i	$0.319580\pi$
$912$	79.5823i	2.63523i
$913$	61.8110i	2.04565i
$914$	−54.8901	−1.81560
$915$	0	0
$916$	106.293	3.51203
$917$	17.6639i	0.583315i
$918$	10.6010i	0.349884i
$919$	−58.0857	−1.91607	−0.958035	−	0.286650i	$-0.907458\pi$
$919$	−58.0857	−1.91607	−0.958035	+	0.286650i	$0.907458\pi$
$920$	0	0
$921$	−0.447506	−0.0147458
$922$	94.2372i	3.10354i
$923$	− 21.3009i	− 0.701127i
$924$	−210.063	−6.91057
$925$	0	0
$926$	91.8991	3.01999
$927$	− 30.6868i	− 1.00789i
$928$	137.821i	4.52419i
$929$	7.38274	0.242220	0.121110	−	0.992639i	$-0.461355\pi$
$929$	7.38274	0.242220	0.121110	+	0.992639i	$0.461355\pi$
$930$	0	0
$931$	−18.2784	−0.599050
$932$	− 25.3032i	− 0.828834i
$933$	− 22.8986i	− 0.749666i
$934$	88.4802	2.89516
$935$	0	0
$936$	−77.2608	−2.52535
$937$	− 34.2781i	− 1.11982i	−0.828555	−	0.559908i	$-0.810837\pi$
$937$	− 34.2781i	− 1.11982i	0.828555	−	0.559908i	$-0.189163\pi$
$938$	− 70.9494i	− 2.31658i
$939$	12.2296	0.399097
$940$	0	0
$941$	8.74328	0.285023	0.142511	−	0.989793i	$-0.454482\pi$
$941$	8.74328	0.285023	0.142511	+	0.989793i	$0.454482\pi$
$942$	− 81.0428i	− 2.64052i
$943$	− 1.17190i	− 0.0381625i
$944$	29.1122	0.947522
$945$	0	0
$946$	−92.8058	−3.01738
$947$	41.2549i	1.34060i	0.742089	+	0.670302i	$0.233835\pi$
$947$	41.2549i	1.34060i	−0.742089	+	0.670302i	$0.766165\pi$
$948$	− 82.2958i	− 2.67284i
$949$	−0.517021	−0.0167832
$950$	0	0
$951$	−56.8912	−1.84482
$952$	− 81.4786i	− 2.64074i
$953$	− 13.2048i	− 0.427745i	−0.976862	−	0.213873i	$-0.931392\pi$
$953$	− 13.2048i	− 0.427745i	0.976862	−	0.213873i	$-0.0686078\pi$
$954$	0.673796	0.0218149
$955$	0	0
$956$	−52.2654	−1.69038
$957$	− 66.7195i	− 2.15673i
$958$	− 52.7835i	− 1.70536i
$959$	54.0861	1.74653
$960$	0	0
$961$	−25.5208	−0.823253
$962$	− 10.0116i	− 0.322785i
$963$	− 15.0399i	− 0.484652i
$964$	−8.65950	−0.278904
$965$	0	0
$966$	6.33976	0.203979
$967$	12.1924i	0.392082i	0.980596	+	0.196041i	$0.0628086\pi$
$967$	12.1924i	0.392082i	−0.980596	+	0.196041i	$0.937191\pi$
$968$	− 79.3131i	− 2.54922i
$969$	12.5345	0.402667
$970$	0	0
$971$	6.80253	0.218303	0.109152	−	0.994025i	$-0.465187\pi$
$971$	6.80253	0.218303	0.109152	+	0.994025i	$0.465187\pi$
$972$	104.967i	3.36682i
$973$	− 50.5758i	− 1.62138i
$974$	−6.43214	−0.206099
$975$	0	0
$976$	57.5367	1.84171
$977$	55.4080i	1.77266i	0.463056	+	0.886329i	$0.346753\pi$
$977$	55.4080i	1.77266i	−0.463056	+	0.886329i	$0.653247\pi$
$978$	− 14.5870i	− 0.466442i
$979$	65.4492	2.09177
$980$	0	0
$981$	−10.9528	−0.349695
$982$	38.5308i	1.22957i
$983$	13.4921i	0.430331i	0.976578	+	0.215165i	$0.0690291\pi$
$983$	13.4921i	0.430331i	−0.976578	+	0.215165i	$0.930971\pi$
$984$	−94.3113	−3.00653
$985$	0	0
$986$	41.0212	1.30638
$987$	− 95.9027i	− 3.05262i
$988$	− 47.6065i	− 1.51456i
$989$	2.04575	0.0650510
$990$	0	0
$991$	−2.47900	−0.0787481	−0.0393740	−	0.999225i	$-0.512536\pi$
$991$	−2.47900	−0.0787481	−0.0393740	+	0.999225i	$0.512536\pi$
$992$	48.9839i	1.55524i
$993$	− 36.4489i	− 1.15667i
$994$	60.4029	1.91587
$995$	0	0
$996$	−173.811	−5.50741
$997$	− 13.6685i	− 0.432885i	−0.976295	−	0.216443i	$-0.930555\pi$
$997$	− 13.6685i	− 0.432885i	0.976295	−	0.216443i	$-0.0694455\pi$
$998$	84.8534i	2.68599i
$999$	−1.70198	−0.0538484

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	925.2.b.f.149.10		10
5.2	odd	4		925.2.a.f.1.1		5
5.3	odd	4		185.2.a.e.1.5	✓	5
5.4	even	2	inner	925.2.b.f.149.1		10
15.2	even	4		8325.2.a.ch.1.5		5
15.8	even	4		1665.2.a.p.1.1		5
20.3	even	4		2960.2.a.w.1.5		5
35.13	even	4		9065.2.a.k.1.5		5
185.73	odd	4		6845.2.a.f.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.a.e.1.5	✓	5	5.3	odd	4
925.2.a.f.1.1		5	5.2	odd	4
925.2.b.f.149.1		10	5.4	even	2	inner
925.2.b.f.149.10		10	1.1	even	1	trivial
1665.2.a.p.1.1		5	15.8	even	4
2960.2.a.w.1.5		5	20.3	even	4
6845.2.a.f.1.1		5	185.73	odd	4
8325.2.a.ch.1.5		5	15.2	even	4
9065.2.a.k.1.5		5	35.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 925 = 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	925.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.72362i q^{2} +2.29298i q^{3} -5.41809 q^{4} -6.24519 q^{6} +3.82710i q^{7} -9.30957i q^{8} -2.25774 q^{9} -4.41809 q^{11} -12.4236i q^{12} +3.67583i q^{13} -10.4236 q^{14} +14.5195 q^{16} -2.28688i q^{17} -6.14922i q^{18} +2.39037 q^{19} -8.77545 q^{21} -12.0332i q^{22} +0.265251i q^{23} +21.3466 q^{24} -10.0116 q^{26} +1.70198i q^{27} -20.7356i q^{28} +6.58595 q^{29} +2.34076 q^{31} +20.9265i q^{32} -10.1306i q^{33} +6.22860 q^{34} +12.2326 q^{36} +1.00000i q^{37} +6.51044i q^{38} -8.42859 q^{39} -4.41809 q^{41} -23.9010i q^{42} -7.71249i q^{43} +23.9376 q^{44} -0.722443 q^{46} +10.9285i q^{47} +33.2929i q^{48} -7.64669 q^{49} +5.24377 q^{51} -19.9160i q^{52} +0.109574i q^{53} -4.63555 q^{54} +35.6286 q^{56} +5.48105i q^{57} +17.9376i q^{58} +2.00504 q^{59} +3.96271 q^{61} +6.37534i q^{62} -8.64059i q^{63} -27.9567 q^{64} +27.5918 q^{66} +6.80664i q^{67} +12.3905i q^{68} -0.608215 q^{69} -5.79485 q^{71} +21.0186i q^{72} +0.140654i q^{73} -2.72362 q^{74} -12.9512 q^{76} -16.9085i q^{77} -22.9563i q^{78} +6.62418 q^{79} -10.6758 q^{81} -12.0332i q^{82} -13.9904i q^{83} +47.5462 q^{84} +21.0059 q^{86} +15.1014i q^{87} +41.1305i q^{88} -14.8139 q^{89} -14.0678 q^{91} -1.43716i q^{92} +5.36731i q^{93} -29.7651 q^{94} -47.9839 q^{96} -8.94394i q^{97} -20.8266i q^{98} +9.97490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 20 q^{4} - 12 q^{6} - 12 q^{9} - 10 q^{11} + 16 q^{14} + 32 q^{16} + 8 q^{19} + 6 q^{21} + 84 q^{24} - 8 q^{26} + 8 q^{29} + 16 q^{31} + 64 q^{34} + 32 q^{36} - 4 q^{39} - 10 q^{41} + 92 q^{44} - 44 q^{49}+ \cdots + 20 q^{99}+O(q^{100}) \) 10 * q - 20 * q^4 - 12 * q^6 - 12 * q^9 - 10 * q^11 + 16 * q^14 + 32 * q^16 + 8 * q^19 + 6 * q^21 + 84 * q^24 - 8 * q^26 + 8 * q^29 + 16 * q^31 + 64 * q^34 + 32 * q^36 - 4 * q^39 - 10 * q^41 + 92 * q^44 - 44 * q^49 - 4 * q^51 + 20 * q^54 + 16 * q^56 + 60 * q^59 - 28 * q^61 - 40 * q^64 + 96 * q^66 - 16 * q^69 - 14 * q^71 - 4 * q^74 + 24 * q^76 - 56 * q^79 - 62 * q^81 + 36 * q^84 + 88 * q^86 - 12 * q^89 - 28 * q^91 - 8 * q^94 - 84 * q^96 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more